STATISTICAL PHYSICS EXAM 2026 WITH
VERIFIED QUESTIONS AND ANSWERS
How does the extension of a polymer scale with chain length when stretched by
a Stokes drag force - ANSWER-Balancing the entropic spring force against
drag gives r ∝ N b² R v / (k_B T). For an ideal chain R ~ N^{1/2}, giving r ~
N^{3/2}.
How does Stokes-drag-induced extension scale for a self-avoiding polymer -
ANSWER-The hydrodynamic radius scales as R ~ N^{3/5} and the spring
constant scales as K ~ N^{−6/5}. Balancing drag against this softer spring gives
r ~ N^{9/5}.
What is Flory theory - ANSWER-A scaling argument estimating polymer size
with interactions. It balances entropic stretching against repulsive monomer-
monomer interactions.
Why does Flory theory predict a swollen chain - ANSWER-Repulsive
interactions raise the energy of compact configurations. The chain expands
beyond ideal size to dilute monomer density.
What is the self-avoiding walk - ANSWER-A random walk forbidden from
revisiting any site. It models a polymer with hard-core excluded volume
between monomers and gives the Flory exponent ν ≈ 3/5 in three dimensions.
What is the persistence length - ANSWER-The distance along a polymer over
which bond orientations remain correlated. Bonds further apart than this lose
memory of each other's directions.
,What characterises a flexible polymer - ANSWER-A contour length much
longer than its persistence length. The chain explores many independent
orientations and behaves as a random walk on large scales.
What characterises a stiff polymer - ANSWER-A contour length comparable to
or shorter than its persistence length. Bond orientations remain correlated along
most of the chain, suppressing random-walk behaviour.
What is the theta condition - ANSWER-The temperature or solvent quality at
which the second virial coefficient vanishes. The chain then behaves as if non-
interacting and recovers ideal scaling.
Why is the central limit theorem relevant to polymer statistics - ANSWER-The
end-to-end vector is a sum of many independent bond vectors. Its distribution
becomes Gaussian for large N regardless of the single-bond distribution.
Why is the entropy of an ideal gas extensive - ANSWER-Including the
indistinguishability factor N! in the partition function makes ln Z proportional
to N at fixed density. Without it the entropy would grow super-linearly and
violate thermodynamics.
What is the microcanonical ensemble - ANSWER-A description of an isolated
system at fixed energy, volume, and particle number. All accessible microstates
with the prescribed energy are taken to be equally probable.
Why does the heat capacity diverge or jump at a continuous transition -
ANSWER-Ordering near T_c absorbs energy as the order parameter rearranges,
on top of the usual heat capacity. The contribution to ∂E/∂T is singular because
the response time and amplitude both grow.
Why is rubber's stress-strain non-Hookean at finite strain - ANSWER-Each
subchain's force-extension is itself non-linear away from small extensions.
Affine deformation transmits this non-linearity to macroscopic stress.
, Why is the demixing tendency captured by a single parameter Δ - ANSWER-
Only the difference between unlike-pair and like-pair interaction energies
matters for mixing. The like-pair contributions act as a constant background.
What physical mechanism causes phase separation in real fluids - ANSWER-
Differences in molecular interactions between unlike and like species. When
unlike contacts are unfavourable enough, energy savings from segregation
overcome the entropy cost.
Why is nucleation needed in the metastable region - ANSWER-The free energy
has only a local minimum at the homogeneous composition. A finite-amplitude
droplet of the favoured phase must form before phase separation can proceed.
What is the leading-order paramagnetic susceptibility from Landau theory -
ANSWER-Above T_c, free energy minimisation gives m ≈ H / [a(T − T_c)].
The susceptibility ∂m/∂H ~ 1/(T − T_c), the Curie-Weiss law.
What is the spontaneous magnetisation in Landau theory below T_c -
ANSWER-Setting H = 0 and minimising the quartic free energy gives m =
±√[a(T_c − T)/u]. The square-root vanishing is the mean field exponent β = 1/2.
How does Landau theory determine the direction of a vector magnetisation -
ANSWER-With an applied field, the term −H·m forces m to align with H. With
no field, all directions are degenerate and one is selected by spontaneous
symmetry breaking.
Why does the dimensionless ratio of the Landau magnetisations suggest scaling
- ANSWER-The ratio H/[a(T−T_c)] divided by √[a|T−T_c|/u] gives a
dimensionless combination involving |T−T_c|^{3/2}. Its appearance suggests m
/ |T−T_c|^{1/2} is a function of this single combination, i.e. a scaling form.
VERIFIED QUESTIONS AND ANSWERS
How does the extension of a polymer scale with chain length when stretched by
a Stokes drag force - ANSWER-Balancing the entropic spring force against
drag gives r ∝ N b² R v / (k_B T). For an ideal chain R ~ N^{1/2}, giving r ~
N^{3/2}.
How does Stokes-drag-induced extension scale for a self-avoiding polymer -
ANSWER-The hydrodynamic radius scales as R ~ N^{3/5} and the spring
constant scales as K ~ N^{−6/5}. Balancing drag against this softer spring gives
r ~ N^{9/5}.
What is Flory theory - ANSWER-A scaling argument estimating polymer size
with interactions. It balances entropic stretching against repulsive monomer-
monomer interactions.
Why does Flory theory predict a swollen chain - ANSWER-Repulsive
interactions raise the energy of compact configurations. The chain expands
beyond ideal size to dilute monomer density.
What is the self-avoiding walk - ANSWER-A random walk forbidden from
revisiting any site. It models a polymer with hard-core excluded volume
between monomers and gives the Flory exponent ν ≈ 3/5 in three dimensions.
What is the persistence length - ANSWER-The distance along a polymer over
which bond orientations remain correlated. Bonds further apart than this lose
memory of each other's directions.
,What characterises a flexible polymer - ANSWER-A contour length much
longer than its persistence length. The chain explores many independent
orientations and behaves as a random walk on large scales.
What characterises a stiff polymer - ANSWER-A contour length comparable to
or shorter than its persistence length. Bond orientations remain correlated along
most of the chain, suppressing random-walk behaviour.
What is the theta condition - ANSWER-The temperature or solvent quality at
which the second virial coefficient vanishes. The chain then behaves as if non-
interacting and recovers ideal scaling.
Why is the central limit theorem relevant to polymer statistics - ANSWER-The
end-to-end vector is a sum of many independent bond vectors. Its distribution
becomes Gaussian for large N regardless of the single-bond distribution.
Why is the entropy of an ideal gas extensive - ANSWER-Including the
indistinguishability factor N! in the partition function makes ln Z proportional
to N at fixed density. Without it the entropy would grow super-linearly and
violate thermodynamics.
What is the microcanonical ensemble - ANSWER-A description of an isolated
system at fixed energy, volume, and particle number. All accessible microstates
with the prescribed energy are taken to be equally probable.
Why does the heat capacity diverge or jump at a continuous transition -
ANSWER-Ordering near T_c absorbs energy as the order parameter rearranges,
on top of the usual heat capacity. The contribution to ∂E/∂T is singular because
the response time and amplitude both grow.
Why is rubber's stress-strain non-Hookean at finite strain - ANSWER-Each
subchain's force-extension is itself non-linear away from small extensions.
Affine deformation transmits this non-linearity to macroscopic stress.
, Why is the demixing tendency captured by a single parameter Δ - ANSWER-
Only the difference between unlike-pair and like-pair interaction energies
matters for mixing. The like-pair contributions act as a constant background.
What physical mechanism causes phase separation in real fluids - ANSWER-
Differences in molecular interactions between unlike and like species. When
unlike contacts are unfavourable enough, energy savings from segregation
overcome the entropy cost.
Why is nucleation needed in the metastable region - ANSWER-The free energy
has only a local minimum at the homogeneous composition. A finite-amplitude
droplet of the favoured phase must form before phase separation can proceed.
What is the leading-order paramagnetic susceptibility from Landau theory -
ANSWER-Above T_c, free energy minimisation gives m ≈ H / [a(T − T_c)].
The susceptibility ∂m/∂H ~ 1/(T − T_c), the Curie-Weiss law.
What is the spontaneous magnetisation in Landau theory below T_c -
ANSWER-Setting H = 0 and minimising the quartic free energy gives m =
±√[a(T_c − T)/u]. The square-root vanishing is the mean field exponent β = 1/2.
How does Landau theory determine the direction of a vector magnetisation -
ANSWER-With an applied field, the term −H·m forces m to align with H. With
no field, all directions are degenerate and one is selected by spontaneous
symmetry breaking.
Why does the dimensionless ratio of the Landau magnetisations suggest scaling
- ANSWER-The ratio H/[a(T−T_c)] divided by √[a|T−T_c|/u] gives a
dimensionless combination involving |T−T_c|^{3/2}. Its appearance suggests m
/ |T−T_c|^{1/2} is a function of this single combination, i.e. a scaling form.
